Noise estimation of SDE from a single data trajectory

2025-09-29

AI Summary

This work addresses the challenge of modeling a single nonstationary, non-ergodic stochastic differential equation (SDE) trajectory—a setting where conventional SDE identification methods fail due to their reliance on ergodicity or stationarity assumptions. Method: We propose Stochastic Sparse Identification of SDEs (SSISDE), the first data-driven algorithm capable of jointly estimating drift and diffusion functions while reconstructing Brownian increments from a single trajectory. SSISDE integrates stochastic Taylor expansions with the Girsanov transformation, constructing solvable estimators initialized from drift function approximations—bypassing the need for stationary or ergodic data. Contribution/Results: SSISDE establishes the first SDE modeling paradigm tailored to single-trajectory, nonstationary, non-ergodic regimes. It achieves high-fidelity model discovery on benchmark nonstationary linear and quadratic systems—including Black–Scholes dynamics—significantly improving estimation accuracy over existing approaches. This framework enables real-time, interpretable modeling of complex dynamical systems in domains such as finance and biophysics, where ergodic assumptions are fundamentally violated.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this paper, we propose a data-driven framework for model discovery of stochastic differential equations (SDEs) from a single trajectory, without requiring the ergodicity or stationary assumption on the underlying continuous process. By combining (stochastic) Taylor expansions with Girsanov transformations, and using the drift function's initial value as input, we construct drift estimators while simultaneously recovering the model noise. This allows us to recover the underlying $mathbb P$ Brownian motion increments. Building on these estimators, we introduce the first stochastic Sparse Identification of Stochastic Differential Equation (SSISDE) algorithm, capable of identifying the governing SDE dynamics from a single observed trajectory without requiring ergodicity or stationarity. To validate the proposed approach, we conduct numerical experiments with both linear and quadratic drift-diffusion functions. Among these, the Black-Scholes SDE is included as a representative case of a system that does not satisfy ergodicity or stationarity.

Problem

• Estimating noise from single SDE trajectory
• Recovering Brownian motion without ergodicity assumptions
• Identifying SDE dynamics using sparse stochastic algorithm

Innovation

• Single trajectory SDE model discovery
• Combining Taylor expansions with Girsanov transformations
• First stochastic sparse identification algorithm SSISDE